Abstract

We obtain analytical conditions for the occurrence of numerical instability (NI) of a split-step method when the simulated solution of the nonlinear Schrödinger equation is close to a plane wave with nonzero carrier frequency. We also numerically study such an instability when the solution is a sequence of pulses rather than a plane wave. The plane-wave-based analysis gives reasonable predictions for the frequencies of the numerically unstable Fourier modes but overestimates the instability growth rate. The latter is found to be strongly influenced by the randomness of the signal’s profile: The more randomly it varies during the propagation, the weaker is the NI. Using an example of a realistic transmission system, we demonstrate that our single-channel results can be used to predict occurrences of NI in multichannel simulations. We also give an estimate for the integration step size for which NI, while present, will not affect simulation results for such systems. Using that estimate may lead to a significant saving of computational time.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).
  2. G. Strang, “On the construction and comparison of difference schemes,” SIAM J. Numer. Anal. 5, 506–517 (1968).
    [Crossref]
  3. M. Glassner, D. Yevick, and B. Hermansson, “High-order generalized propagation techniques,” J. Opt. Soc. Am. B 8, 413–415 (1991).
    [Crossref]
  4. A. A. Rieznik, T. Tolisano, F. A. Callegari, D. F. Grosz, and H. L. Fragnito, “Uncertainty relation for the optimization of optical-fiber transmission systems simulations,” Opt. Express 13, 3822–3834 (2005).
    [Crossref]
  5. Q. Zhang and M. I. Hayee, “Symmetrized split-step Fourier scheme to control global simulation accuracy in fiber-optic communication systems,” J. Lightwave Technol. 26, 302–316 (2008).
    [Crossref]
  6. T. I. Lakoba, “Instability analysis of the split-step Fourier method on the background of a soliton of the nonlinear Schrödinger equation,” Num. Meth. Part. Diff. Eqs. 28, 641–669 (2012).
  7. T. I. Lakoba, “Instability of the finite-difference split-step method on the background of a soliton of the nonlinear Schrödinger equation,” http://arxiv.org/abs/1208.0578v1 .
  8. G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Suppression of spurious tones induced by the split-step method in fiber systems simulations,” IEEE Photon. Technol. Lett. 12, 489–491 (2000).
    [Crossref]
  9. G. Li, “Recent advances in coherent optical communication,” Adv. Opt. Photon. 1, 279–307 (2009).
    [Crossref]
  10. X. Li, X. Chen, and M. Qasmi, “A broad-band digital filtering approach for time-domain simulation of pulse propagation in optical fiber,” J. Lightw. Technol. 23, 864–875 (2005).
    [Crossref]
  11. T. Kremp, “Split-step quasi-spectral finite difference method for nonlinear optical pulse propagation,” in Optical Fiber Communication Conference, OSA Technical Digest Series (Optical Society of America, 2006), paper OWI8.
  12. P. G. Kevrekidis, The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numerical Computations and Physical Perspectives (Springer, 2009).
  13. J. A. C. Weideman and B. M. Herbst, “Split-step methods for the solution of the nonlinear Schrödinger equation,” SIAM J. Numer. Anal. 23, 485–507 (1986).
    [Crossref]
  14. F. Matera, A. Mecozzi, M. Romagnoli, and M. Settembre, “Sideband instability induced by periodic power variation in long-distance fiber links,” Opt. Lett. 18, 1499–1501 (1993).
    [Crossref]
  15. S. A. Chin, “Higher-order splitting algorithms for solving the nonlinear Schrödinger equation and their instabilities,” Phys. Rev. A 76, 056708 (2007).
  16. E. Faou, L. Gauckler, and C. Lubich, “Plane wave stability of the split-step Fourier method for the nonlinear Schrödinger equation,” http://arxiv.org/abs/1306.0656v1 .
  17. B. Cano and A. Gonzalez-Pachon, “Plane waves numerical stability of some explicit exponential methods for cubic Schrödinger equation,” preprint: Appl. Math. Report, (Univ. Valladolid, 2013).
  18. M. H. Holmes, Introduction to Perturbation Methods (Springer, 1995); Sec. 1.5.
  19. D. F. Grosz, A. Agarwal, S. Banerjee, D. N. Maywar, and A. P. Küng, “All-Raman ultra-long-haul single wideband DWDM transmission systems with OADM capability,” J. Lightwave Technol. 22, 423–432 (2004).
    [Crossref]
  20. D. A. Fishman, W. A. Thompson, and L. Vallone, “LambdaXtreme transport system: R&D of a high capacity system for low cost, ultra long haul DWDM transport,” Bell Labs Techn. J. 11, 27–53 (2006).
  21. T. I. Lakoba, “Transmission improvement in ultralong dispersion-managed soliton WDM systems by using pulses with different widths,” J. Lightwave Technol. 23, 2647–2653 (2005).
    [Crossref]
  22. S. Derevyanko, “The (n+1)/2 rule for dealiasing in the split-step Fourier methods for n-wave interactions,” IEEE Photon. Technol. Lett. 20, 1929–1931 (2008).
    [Crossref]
  23. J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed. (Dover, 2001), Sec. 11.5.

2012 (1)

T. I. Lakoba, “Instability analysis of the split-step Fourier method on the background of a soliton of the nonlinear Schrödinger equation,” Num. Meth. Part. Diff. Eqs. 28, 641–669 (2012).

2009 (1)

2008 (2)

Q. Zhang and M. I. Hayee, “Symmetrized split-step Fourier scheme to control global simulation accuracy in fiber-optic communication systems,” J. Lightwave Technol. 26, 302–316 (2008).
[Crossref]

S. Derevyanko, “The (n+1)/2 rule for dealiasing in the split-step Fourier methods for n-wave interactions,” IEEE Photon. Technol. Lett. 20, 1929–1931 (2008).
[Crossref]

2007 (1)

S. A. Chin, “Higher-order splitting algorithms for solving the nonlinear Schrödinger equation and their instabilities,” Phys. Rev. A 76, 056708 (2007).

2006 (1)

D. A. Fishman, W. A. Thompson, and L. Vallone, “LambdaXtreme transport system: R&D of a high capacity system for low cost, ultra long haul DWDM transport,” Bell Labs Techn. J. 11, 27–53 (2006).

2005 (3)

2004 (1)

2000 (1)

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Suppression of spurious tones induced by the split-step method in fiber systems simulations,” IEEE Photon. Technol. Lett. 12, 489–491 (2000).
[Crossref]

1993 (1)

1991 (1)

1986 (1)

J. A. C. Weideman and B. M. Herbst, “Split-step methods for the solution of the nonlinear Schrödinger equation,” SIAM J. Numer. Anal. 23, 485–507 (1986).
[Crossref]

1968 (1)

G. Strang, “On the construction and comparison of difference schemes,” SIAM J. Numer. Anal. 5, 506–517 (1968).
[Crossref]

Agarwal, A.

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

Banerjee, S.

Benedetto, S.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Suppression of spurious tones induced by the split-step method in fiber systems simulations,” IEEE Photon. Technol. Lett. 12, 489–491 (2000).
[Crossref]

Bosco, G.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Suppression of spurious tones induced by the split-step method in fiber systems simulations,” IEEE Photon. Technol. Lett. 12, 489–491 (2000).
[Crossref]

Boyd, J. P.

J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed. (Dover, 2001), Sec. 11.5.

Callegari, F. A.

Cano, B.

B. Cano and A. Gonzalez-Pachon, “Plane waves numerical stability of some explicit exponential methods for cubic Schrödinger equation,” preprint: Appl. Math. Report, (Univ. Valladolid, 2013).

Carena, A.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Suppression of spurious tones induced by the split-step method in fiber systems simulations,” IEEE Photon. Technol. Lett. 12, 489–491 (2000).
[Crossref]

Chen, X.

X. Li, X. Chen, and M. Qasmi, “A broad-band digital filtering approach for time-domain simulation of pulse propagation in optical fiber,” J. Lightw. Technol. 23, 864–875 (2005).
[Crossref]

Chin, S. A.

S. A. Chin, “Higher-order splitting algorithms for solving the nonlinear Schrödinger equation and their instabilities,” Phys. Rev. A 76, 056708 (2007).

Curri, V.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Suppression of spurious tones induced by the split-step method in fiber systems simulations,” IEEE Photon. Technol. Lett. 12, 489–491 (2000).
[Crossref]

Derevyanko, S.

S. Derevyanko, “The (n+1)/2 rule for dealiasing in the split-step Fourier methods for n-wave interactions,” IEEE Photon. Technol. Lett. 20, 1929–1931 (2008).
[Crossref]

Faou, E.

E. Faou, L. Gauckler, and C. Lubich, “Plane wave stability of the split-step Fourier method for the nonlinear Schrödinger equation,” http://arxiv.org/abs/1306.0656v1 .

Fishman, D. A.

D. A. Fishman, W. A. Thompson, and L. Vallone, “LambdaXtreme transport system: R&D of a high capacity system for low cost, ultra long haul DWDM transport,” Bell Labs Techn. J. 11, 27–53 (2006).

Fragnito, H. L.

Gauckler, L.

E. Faou, L. Gauckler, and C. Lubich, “Plane wave stability of the split-step Fourier method for the nonlinear Schrödinger equation,” http://arxiv.org/abs/1306.0656v1 .

Gaudino, R.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Suppression of spurious tones induced by the split-step method in fiber systems simulations,” IEEE Photon. Technol. Lett. 12, 489–491 (2000).
[Crossref]

Glassner, M.

Gonzalez-Pachon, A.

B. Cano and A. Gonzalez-Pachon, “Plane waves numerical stability of some explicit exponential methods for cubic Schrödinger equation,” preprint: Appl. Math. Report, (Univ. Valladolid, 2013).

Grosz, D. F.

Hayee, M. I.

Herbst, B. M.

J. A. C. Weideman and B. M. Herbst, “Split-step methods for the solution of the nonlinear Schrödinger equation,” SIAM J. Numer. Anal. 23, 485–507 (1986).
[Crossref]

Hermansson, B.

Holmes, M. H.

M. H. Holmes, Introduction to Perturbation Methods (Springer, 1995); Sec. 1.5.

Kevrekidis, P. G.

P. G. Kevrekidis, The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numerical Computations and Physical Perspectives (Springer, 2009).

Kremp, T.

T. Kremp, “Split-step quasi-spectral finite difference method for nonlinear optical pulse propagation,” in Optical Fiber Communication Conference, OSA Technical Digest Series (Optical Society of America, 2006), paper OWI8.

Küng, A. P.

Lakoba, T. I.

T. I. Lakoba, “Instability analysis of the split-step Fourier method on the background of a soliton of the nonlinear Schrödinger equation,” Num. Meth. Part. Diff. Eqs. 28, 641–669 (2012).

T. I. Lakoba, “Transmission improvement in ultralong dispersion-managed soliton WDM systems by using pulses with different widths,” J. Lightwave Technol. 23, 2647–2653 (2005).
[Crossref]

T. I. Lakoba, “Instability of the finite-difference split-step method on the background of a soliton of the nonlinear Schrödinger equation,” http://arxiv.org/abs/1208.0578v1 .

Li, G.

Li, X.

X. Li, X. Chen, and M. Qasmi, “A broad-band digital filtering approach for time-domain simulation of pulse propagation in optical fiber,” J. Lightw. Technol. 23, 864–875 (2005).
[Crossref]

Lubich, C.

E. Faou, L. Gauckler, and C. Lubich, “Plane wave stability of the split-step Fourier method for the nonlinear Schrödinger equation,” http://arxiv.org/abs/1306.0656v1 .

Matera, F.

Maywar, D. N.

Mecozzi, A.

Poggiolini, P.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Suppression of spurious tones induced by the split-step method in fiber systems simulations,” IEEE Photon. Technol. Lett. 12, 489–491 (2000).
[Crossref]

Qasmi, M.

X. Li, X. Chen, and M. Qasmi, “A broad-band digital filtering approach for time-domain simulation of pulse propagation in optical fiber,” J. Lightw. Technol. 23, 864–875 (2005).
[Crossref]

Rieznik, A. A.

Romagnoli, M.

Settembre, M.

Strang, G.

G. Strang, “On the construction and comparison of difference schemes,” SIAM J. Numer. Anal. 5, 506–517 (1968).
[Crossref]

Thompson, W. A.

D. A. Fishman, W. A. Thompson, and L. Vallone, “LambdaXtreme transport system: R&D of a high capacity system for low cost, ultra long haul DWDM transport,” Bell Labs Techn. J. 11, 27–53 (2006).

Tolisano, T.

Vallone, L.

D. A. Fishman, W. A. Thompson, and L. Vallone, “LambdaXtreme transport system: R&D of a high capacity system for low cost, ultra long haul DWDM transport,” Bell Labs Techn. J. 11, 27–53 (2006).

Weideman, J. A. C.

J. A. C. Weideman and B. M. Herbst, “Split-step methods for the solution of the nonlinear Schrödinger equation,” SIAM J. Numer. Anal. 23, 485–507 (1986).
[Crossref]

Yevick, D.

Zhang, Q.

Adv. Opt. Photon. (1)

Bell Labs Techn. J. (1)

D. A. Fishman, W. A. Thompson, and L. Vallone, “LambdaXtreme transport system: R&D of a high capacity system for low cost, ultra long haul DWDM transport,” Bell Labs Techn. J. 11, 27–53 (2006).

IEEE Photon. Technol. Lett. (2)

S. Derevyanko, “The (n+1)/2 rule for dealiasing in the split-step Fourier methods for n-wave interactions,” IEEE Photon. Technol. Lett. 20, 1929–1931 (2008).
[Crossref]

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Suppression of spurious tones induced by the split-step method in fiber systems simulations,” IEEE Photon. Technol. Lett. 12, 489–491 (2000).
[Crossref]

J. Lightw. Technol. (1)

X. Li, X. Chen, and M. Qasmi, “A broad-band digital filtering approach for time-domain simulation of pulse propagation in optical fiber,” J. Lightw. Technol. 23, 864–875 (2005).
[Crossref]

J. Lightwave Technol. (3)

J. Opt. Soc. Am. B (1)

Num. Meth. Part. Diff. Eqs. (1)

T. I. Lakoba, “Instability analysis of the split-step Fourier method on the background of a soliton of the nonlinear Schrödinger equation,” Num. Meth. Part. Diff. Eqs. 28, 641–669 (2012).

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (1)

S. A. Chin, “Higher-order splitting algorithms for solving the nonlinear Schrödinger equation and their instabilities,” Phys. Rev. A 76, 056708 (2007).

SIAM J. Numer. Anal. (2)

J. A. C. Weideman and B. M. Herbst, “Split-step methods for the solution of the nonlinear Schrödinger equation,” SIAM J. Numer. Anal. 23, 485–507 (1986).
[Crossref]

G. Strang, “On the construction and comparison of difference schemes,” SIAM J. Numer. Anal. 5, 506–517 (1968).
[Crossref]

Other (8)

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

T. Kremp, “Split-step quasi-spectral finite difference method for nonlinear optical pulse propagation,” in Optical Fiber Communication Conference, OSA Technical Digest Series (Optical Society of America, 2006), paper OWI8.

P. G. Kevrekidis, The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numerical Computations and Physical Perspectives (Springer, 2009).

T. I. Lakoba, “Instability of the finite-difference split-step method on the background of a soliton of the nonlinear Schrödinger equation,” http://arxiv.org/abs/1208.0578v1 .

E. Faou, L. Gauckler, and C. Lubich, “Plane wave stability of the split-step Fourier method for the nonlinear Schrödinger equation,” http://arxiv.org/abs/1306.0656v1 .

B. Cano and A. Gonzalez-Pachon, “Plane waves numerical stability of some explicit exponential methods for cubic Schrödinger equation,” preprint: Appl. Math. Report, (Univ. Valladolid, 2013).

M. H. Holmes, Introduction to Perturbation Methods (Springer, 1995); Sec. 1.5.

J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed. (Dover, 2001), Sec. 11.5.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1.
Fig. 1.

Comparison of Eqs. (5) and (9).

Fig. 2.
Fig. 2.

(a) Growth rate of unstable modes for Δz=1.25Δzthr and Ω0=ωmax/3: β2/2=1 (solid red), β2/2=1 (dashed black); modulationally unstable modes are shown by dotted black line. (b) δ(ω) for Δz=1.25Δzthr, β2/2=1, β3=0, and two values of Ω0; other notations are explained in the text. (c) Similar to (b) with Ω0=ωmax/3, but for β3=±0.4β2/ωmax. Note that δ(ω) is discontinuous.

Fig. 3.
Fig. 3.

(a) Centers of the instability bands, given by Eq. (26), for Ω0/ωmax=0.2 (blue, circle), 0.3 (red, square), and 0.4 (black). Note that Δzthr depend on Ω0. Thin and thick lines are for N=1 and N=2, respectively. Solid and dashed lines are for the instability peaks in the main and aliased intervals, respectively. (b) Ratios of the widths of the same instability bands as in (a). (c) Centers of the instability bands, given by Eq. (35), for Ω0/ωmax=0.3 and β3ωmax/β2=0.4 (red) and 0.4 (blue). Line thicknesses and styles denote the same as in (a). The pointing arrows emphasize that the curves are discontinuous at ω=ωmax+2Ω0.

Fig. 4.
Fig. 4.

(a) Frequencies of the unstable modes in the aliased frequency interval (see Fig. 2): solid line—computed from (26b); circles—obtained numerically as explained in the text. Several circles for the same value of Δz indicate that for this Δz, multiple isolated unstable modes are observed, as illustrated in the next panel. (b) Spectrum of the numerically computed solution for Δz=1.25Δzthr, showing multiple unstable modes in the aliased frequency interval. The vertical dashed line separates the aliased and main frequency intervals.

Fig. 5.
Fig. 5.

Schematics illustrating Facts 1 to 3.

Fig. 6.
Fig. 6.

Simulated signal quality versus Δz.

Fig. 7.
Fig. 7.

Fluctuations of simulated signal quality, defined as max|EC4±nEC4±n,benchmark|, where n=0,,3 and ECk,benchmark is measured for ωmax/(2π)=640GHz and Δz<Δzthr,0 to avoid aliasing and NI. Colors and line styles correspond to the same channels as in Fig. 6. The vertical line marks where Ω0,edge=0.5ωmax. The fluctuation outside the vertical limits of the figure is approximately 6 dB.

Equations (63)

Equations on this page are rendered with MathJax. Learn more.

iuz(β2/2)utti(β3/6)uttt+γu|u|2=0.
fornfrom1tonmaxdo:u¯(t)=un(t)exp[iγ|un(t)|2Δz](nonlinear step)un+1(t)={solution ofiuz=(β2/2)utt+i(β3/6)utttatz=Δzwith initial conditionu¯(t)(dispersive step)
upw=Aexp[iΩ0tiK0z],K0=(β2Ω02/2β3Ω03/6),
un+1(t)=F1[exp[iP(ω)]F[u¯(t)]],
P(ω)=(β2ω2/2β3ω3/6)Δz.
ωmaxωωmax,ωmax=π/Δt,
iun+1mu¯mΔz=β2(un+1m+12un+1m+un+1m1Δt2+u¯m+12u¯m+u¯m1Δt2),
u(T/2,z)=u(T/2,z).
P(ω)=2arctan[β2rsin2(ωΔt/2)],r=Δz/Δt2.
un=uB+u˜n,|u˜n||uB|,
F[u˜n+1]=eiP(ω)F[eiγ|uB|2Δz(u˜n+iγΔz(uB2u˜n*+|uB|2u˜n))].
Ω0>0
ϵγA2Δz1.
u˜n=eiΩ0t+iP(Ω0)n(pneiω1t+qn*eiω^1t).
Ω0ω^1={2Ω0ωif2Ω0ωωmax;2Ω0ω2ωmaxif2Ω0ω>ωmax.
(pn+1qn+1)=(eiΔ+(1+iϵ)eiΔ+iϵeiΔiϵeiΔ(1iϵ))(pnqn),
Δ+=P(Ω0+ω1)P(Ω0),Δ=P(Ω0ω^1)P(Ω0).
λ=exp[iΔ+Δ2](cos(δ+ϵ)cosϵ±(cos(δ+ϵ)cosϵ)21),
δΔ++Δ2=12(P(Ω0+ω1)2P(Ω0)+P(Ω0ω^1)).
Nπ2ϵ<δ<Nπ,NZ.
N=|N|sgn(β2).
P(Ω0+ω1)+P(Ω0ω^1)2P(Ω0)2πN.
δ={β2Δz(ωΩ0)2/2δmainifω(ωmax+2Ω0,ωmax);δali,min+β2Δz(ωΩ0+ωmax)2/2δaliifω[ωmax,ωmax+2Ω0],
δali,min=sgn(β2)πΔzΔzthr(1a2),a=Ω0ωmaxΩ0,
Δzthr=2π/(|β2|(ωmaxΩ0)2).
Ω0<0.5ωmax.
ωNπ,main=Ω0±|N|πsgn(β2)ϵ|β2|Δz/2,
ωNπ,ali=ωmax+Ω0±|N|πsgn(β2)(δali,min+ϵ)|β2|Δz/2,
Δz>|N|Δzthr1+sgn(β2)γA2Δzthr/π,
|N|Δzthr1+sgn(β2)γA2Δzthr/π<Δz<|N|Δzthr1a2+sgn(β2)γA2Δzthr/π,
ωNπ,mainΩ0[1±1a|N|Δzthr/Δz],
ωNπ,ali(ωmaxΩ0)[1±|N|Δzthr/Δz(1a2)],
WNπ,main2ϵ/(|β2|Δz(ωNπ,mainΩ0)),N0;
WNπ,ali2ϵ/(|β2|Δz(ωNπ,ali(ωmax+Ω0))).
ωΩ0>0.5ωmax±=0.5ωmax(1μ±2μ).
β3ωmax<0.5β2,
δ={(β2β3Ω0)Δz(ωΩ0)2/2ifω(ωmax+2Ω0,ωmax);δali,min+(β2+β3(ωmaxΩ0))Δz(ωΩ0+ωmax)2/2ifω[ωmax,ωmax+2Ω0],
δali,min=sgn(β2)πΔzΔzthr(1a2+β3ωmax3β2(1a+a2)),
δjump=β3Δzωmax3/6.
ωNπ,main=Ω0±|N|πsgn(β2)ϵ|β2β3Ω0|Δz/2,
ωNπ,ali=ωmax+Ω0±|N|πsgn(β2)(δali,min+ϵ)|β2+β3(ωmaxΩ0)|Δz/2;
Δz>|N|Δzthr1β3Ω0/β2+sgn(β2)γA2Δzthr/π,
|N|Δzthr1(β3/β2)(Ω0ωmax(1+a)2/3)+sgn(β2)γA2Δzthr/π<Δz<|N|Δzthr1a2+(β3/β2)ωmax(1a+a2)/3+sgn(β2)γA2Δzthr/π,
x3x+2=4x/(β2r)2,x=cos(ωthrΔt).
|β2|r=O(1/(γA2Δz))1/ϵ1.
ωthr2/(31/4|β2|Δz).
Ω00.5ωthr.
max|δ|=π2/(|β2|r)2arctan((Ω0/ωthr)2/3).
2/(β2r)+2arctan((Ω0/ωthr)2/3)<2ϵ.
Δz<Δt/β2γA2(β2Ω0/2)2
Ω02<4γA2/β2.
2ϵ<P(Ω0)(ωΩ0)2/2<0.
0<(ωΩ0)2<2γA2/(β2),
u(t,0)=eiΩ0tj=1Npulsesech[t+(Npulse+12j)Tbit/2]×eiφj+ξ(t),
Δz(X,Y)=Δzthr,0(ωmax/(Ω0,XΩ0,Y))2,
Δzthr,0=2π/(|β2|ωmax2)
ΔzΔzthr,0(ωmax/(Ω0,edge+0.5ΔchΩ))2,
ΔzΔzthr,0(ωmax/(Ω0,edge+1.5ΔchΩ))2,
ω1+ω2+ω3ω4ω5=ω6.
Ω1+Ω1+Ω1Ω2Ω2=(5/3)ωmaxΩ2.
u13u2*2,
Ω1+Ω1+Ω1Ω1Ω2andΩ2+Ω1+Ω1Ω1Ω1,
u12|u1|2u2*and|u1|4u2,

Metrics